On realizing diagrams of Pi-algebras

通过范畴上同调与 incidence 代数的上同调研究图状 Hochschild 上同调

Diagrammatic Hochschild cohomology via cohomology of categories, and incidence algebras

本文通过范畴上同调与 incidence 代数的上同调研究图状 Hochschild 上同调，探讨了 Gerstenhaber-Schack 复合体与 Baues-Wirsching 猜想的联系，并计算了有限滤子 simplicial 复合物的图状 Hochschild 上同调。

代数图的 Hochschild 双复形上的运算

Operations on the Hochschild Bicomplex of a Diagram of Algebras

本文构造了一个作用在代数图 Hochschild 双复形上的 operad，由此直接证明其 Hochschild 上同调为 Gerstenhaber 代数，全复形为 $L_\infty$-代数，并通过 Maurer-Cartan 方程控制形变。

On deformations of diagrams of commutative algebras

In this paper we study classical deformations of diagrams of commutative algebras over a field of characteristic 0. In particular we determine several homotopy classes of DG-Lie algebras, each one of them controlling this above deformation problem: the first homotopy type is described in terms of the projective model structure on the category of diagrams of differential graded algebras, the others in terms of the Reedy model structure on truncated Bousfield-Kan approximations. The first half of the paper contains an elementary introduction to the projective model structure on the category of c…

Laplaza Sets, or How to Select Coherence Diagrams for Pseudo Algebras

We define a general concept of pseudo algebras over theories and 2-theories. A more restrictive such notion was introduced by Hu and Kriz, but as noticed by M. Gould, did not capture the desired examples. The approach taken in this paper corrects the mistake by introducing a more general concept, allowing more flexibility in selecting coherence diagrams for pseudo algebras.

On resolutions of diagrams of algebras

We prove a restricted version of a conjecture by M. Markl on resolutions of an operad describing diagrams of algebras. We discuss a particular case related to the Gerstenhaber-Schack diagram cohomology.

空间的图表的代数与几何最小DG模型

Algebraically and geometrically minimal DG-models of diagrams of spaces

本文推广Sullivan有理同伦理论至有限指标范畴上的空间图表，构造几何与代数最小模型，分类有理同伦类型，并刻画H-图表分裂条件，应用于自同构群计算。

Homotopy Diagrams of Algebras

In [math.AT/9907138] we proved that strongly homotopy algebras are homotopy invariant concepts in the category of chain complexes. Our arguments were based on the fact that strongly homotopy algebras are algebras over minimal cofibrant operads and on the principle that algebras over cofibrant operads are homotopy invariant. In our approach, algebraic models for colored operads describing diagrams of homomorphisms played an important role. The aim of this paper is to give an explicit description of these models. A possible application is an appropriate formulation of the `ideal' homological per…

Natural operations on the Hochschild complex of commutative Frobenius algebras via the complex of looped diagrams

We define a dg-category of looped diagrams which we use to construct operations on the Hochschild complex of commutative Frobenius dg-algebras. We show that we recover the operations known for symmetric Frobenius dg-algebras constructed using Sullivan chord diagrams as well as all formal operations for commutative algebras (including Loday's lambda operations) and prove that there is a chain level version of a suspended Cactus operad inside the complex of looped diagrams. This recovers the suspended BV algebra structure on the Hochschild homology of commutative Frobenius algebras defined by Ab…

广义rook-Brauer代数及其同调

Generalized rook-Brauer algebras and their homology

该研究通过引入带群标记的等变结构与辫子交叉结构推广rook-Brauer代数及其子代数，利用图代数同调方法证明了参数可逆时广义图代数同调与对应群同调的等价性，还得到了奇数边情形下无参数条件的同调等价与同调稳定性结果。

The Steenrod problem of realizing polynomial cohomology rings

In this paper we completely classify which graded polynomial R-algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring R satisfying mild conditions. In the fundamental case R = Z, our result states that the only polynomial cohomology rings over Z which can occur, are tensor products of copies of H^*(CP^\infty;Z) = Z[x_2], H^*(BSU(n);Z) = Z[x_4,x_6,...,x_{2n}], and H^*(BSp(n):Z) = Z[x_4,x_8,...,x_{4n}] confirming an old conjecture. Our classification extends Notbohm's soluti…

The L_\infty-deformation complex of diagrams of algebras

The deformation complex of an algebra over a colored PROP P is defined in terms of a minimal (or, more generally, cofibrant) model of P. It is shown that it carries the structure of an L_\infty-algebra which induces a graded Lie bracket on cohomology. As an example, the L_\infty-algebra structure on the deformation complex of an associative algebra morphism g, with the underlying cochain complex isomorphic to the Gerstenhaber-Schack complex of g, is constructed. Another example is the deformation complex of a Lie algebra morphism. The last example is the diagram describing two mutually inverse…

BV 推出作为拟同构

BV pushforward as a quasi-isomorphism

本文通过同调扰动引理证明 BV 推出映射 P_* 是 BV 复形的拟同构，并给出拟逆映射 i_int 的显式公式。

Deformations associated to rigid algebras

The deformations of an infinite dimensional algebra may be controlled not just by its own cohomology but by that of an associated diagram of algebras, since an infinite dimensional algebra may be absolutely rigid in the classical deformation theory for single algebras while depending essentially on some parameters. Two examples studied here, the function field of a sphere with four marked points and the first Weyl algebra, show, however, that the existence of these parameters may be made evident by the cohomology of a diagram (presheaf) of algebras constructed from the original. The Cohomology…